Folding Paper With Computational Tools

Here is one way to know that your department produced a physics major – a real physics major. A recent graduate sent me two python programs. The first one calculates the value of of Pi to however far you want it to go. The second program calculates the approximate size of paper needed to fold it over a given number of times.

Why did he send me these? Was it for a grade? Clearly, no. He already graduated. Instead, he created these because he was curious. His father had told him he heard about folding paper. Someone had said that if you wanted to fold a piece of paper 50 times, it would have to be as long as the distance from the Earth to the Sun. He wrote a program because he didn’t believe this. Awesome.

Folding Paper

How would you even calculate this size of paper to fold a certain number of times? Here is a nice explanation of the folding paper calculation.

Here is the basic idea. Suppose there is some paper that has a length L and thickness t. Let me show a diagram of the paper after being folded 3 times.

Maybe you should just fold some paper yourself so it is easier to see this. After 3 folds, the paper is essentially 8 times thicker and 1/8th the length of the original paper. For N folds, this gives a thickness to length ratio of:

You can see this ratio explodes rather quickly. The key is that when you fold a paper that is already folded, you double the thickness with each fold and decrease the length by half with each fold. Why look at this ratio at all? Well, eventually the folded thickness will be similar to the folded length. When that happens, you clearly couldn’t fold the paper any more.

Using this folding mathematical model, how many times could you fold an 8.5 x 11 sheet of paper? First, how thick is this paper? That varies, but I already looked at paper before. For plain, multi-use paper I found it to have a thickness of about 10-4 meters per sheet. Of course if you really want to fold some stuff, you could get some thinner paper.

Here is a plot of the thickness to length ratio vs. the number of folds. I have included the plot for the typical 8.5 x 11 sheet as well as a piece of paper that is twice as long and half as thick. Oh, this is for folding in just one direction.

The normal paper reaches the 1 to 1 ratio after 5 folds and the more foldable paper gets you just one more fold. So, you can see how crazy this gets. I really don’t even think a 1 to 1 ratio is feasible for paper folding. I tried as careful as I could to fold plain paper and I just got 4 folds. I could probably squeeze out 5 but it could be questionable whether it was folded or not. For this paper, 4 folds gives a ratio of 0.086 – no where near a ratio of 1.

What If You Want 50 Folds?

This gets back to the question the student was answering. He assumed that you could fold paper as long as the thickness to length ratio was less than 1 (which is just wishful thinking, but ok). Using the ratio equation from before, I can solve for the length:

This is actually greater than the distance from the Earth to the Sun (about 1.5 x 1011 meters). If you used my max folding ratio of 0.086, the distance would be even greater.

Super Size Me

Oh, this wasn’t enough for him. He had to take the problem even further. Here is the output from the python program he wrote.

From this he determined that in order to fold a paper 97 times, it would have to be longer than the visible universe. What do I think is cool about this? He answered the question numerically. You could just algebraically solve for the number of folds, but he didn’t. His program calculates the needed length for each fold. It keeps increasing the number of folds until it gets to the approximate size of the universe. Sure, this might not be the most efficient calculation but that’s ok. The important thing is that it is HIS calculation.

The other cool thing is that he had his go-to tool, python. I’m not saying that python is the only tool anyone should ever use (but maybe that’s true too). Instead I am saying that he had access to a tool. He had it on his computer and he didn’t need a lab manual to guide him through this calculation. I feel pretty comfortable saying that students really need practice at numerical calculations in many of their undergraduate courses in order for a student to get to this level.

Didn’t The MythBusters Do This?

Starting with paper that was 52 meters by 67 meters they were able to fold it 11 times. Now, you need to notice that their folding method is a little different than the above calculation. Their folds alternated directions instead of all being in the same direction. However, the same general idea applies.